Iteration with two analytic fixed points

[bo198214]

To begin, since \(E\) is simply connected, it is safe to assume that \(E = \mathbb{D}\) the unit disk (conjugate with a riemann mapping). A mapping from the unit disk to itself always has a fixed point, and so again, without loss of generality, we can assume one of the fixed points is \(0\). By schwarz's lemma, this means that \(|f'(0)| \le 1\) and that \(|f(z)| \le |z|\). Where \(|f(z)| = |z|\) only when \(|f'(0)| = 1\). This means that \(f(z) = e^{i\theta}z\) for some \(\theta \in [0,2\pi)\).

So, if \(|f'(0)| < 1\), then \(|f(z)| < |z|\) and there are no more fixed points other than \(0\) by construction. So all that's left is if \(f\) has a neutral fixed point at \(0\). Wlog we can assume that \(f'(0) = 1\). Well then \(f(z) = z\), which is only fixed points, and is therefore the trivial iteration. And can be discarded. Or you can just observe that \(e^{i\theta}z\) has no other fixed points than zero.
....

James, I like that you took initiative, BUT:
Your proof has nothing to do with iteration (all your tediously made definition about local iteration find no application in your proof!), it basically states that a holomorphic function \(f\colon E\to E\) can only have one fixed point.
But the assumption \(E\to E\) is already the problem.
One interesting application would be the two fixed points for base sqrt(2).
The theorem should explain why one can not have one iteration holomorphic at both fixed points.
But I have already difficulties to find a domain \(E\) containing both fixed points such that \(b^z\colon E\to E\) (not even started with iteration here) because one of the fixed points is repelling so it tends to map stuff more far away than it came from.
Except perhaps I choose \(E=\mathbb{C}\) (which is simply connected) but then your proof would state that \(b^z\) can only have one fixed point on \(\mathbb{C}\), which is not true - maybe there is problem with "it is safe to assume ... (conjugate Riemann mapping)".

Correct me if I totally misunderstood your statements!